The idiom is "have got" in both American and British English. It is a purely idiomatic alternate for "have" and is not much used except in the present tense. That is, "have got" is the present tense of this idiom, even though "get" is the present tense form of "to get", and the past is "got". This is because "have got" is Present Perfect in form, though not in meaning. To be more specific, "have got" is the British Present Perfect, the American Present Perfect being "have gotten". So in American English we have "My brother has gotten up early this morning", whereas in British English we have "My brother has got up early this morning." Nevertheless, the idiom is "have got" in both AmEng and BrEng, not "have gotten", not even in American English.

I have got a pen. = I have a pen. I have not got any money. = I do not have any money.

Using contractions:

I've got a pen. = I have a pen. I haven't got any money. = I don't have any money.

The use of "have got" for "have" extends to the affirmative form of the semi-modal "have to", so that "have got to" is an idiomatic substitute for "have to" (meaning "must"). The negative form is rare.

I have got to meet my friend at 10 o'clock. = I've got to meet my friend at 10 o'clock. = I have to meet my friend at 10 o'clock.

In fast or less careful speech the contracted "have" ('ve) is glossed over and can barely be heard -- or may not be heard at all. Simultaneously, the "got to" collapses into "gotta". (In American English this is pronounced "godda".) The "z" sound in "he's got" and "she's got" (which stands for "has") remains, however. This leads to the following "conjugation" (using "go" as the complement verb):